In this article, we use methods from fixed point theory to examine a Lambda policy iteration with a randomization algorithm mappings that satisfy the Kannan contraction condition. As shown by examples, this type of mapping extends beyond the scope of the strong contractions considered so far. In particular, we analyze the properties of reinforcement learning methods designed for feedback control, framing our investigation within fixed point theory. Under broad assumptions, we establish sufficient criteria for convergence in policy spaces of infinite dimensions.

We investigate the challenge of designing robust external excitations to control and synchronize a population of non-interacting homotypic harmonic oscillators, specifically, theta neurons. The Theta model emulates the bursting behavior observed in spiking cells, characterized by periodic oscillations in their membrane electric potential.Our approach involves formulating this optimization task as an optimal mean-field control problem for the linear continuity/Fokker-Planck equation in the space of probability measures. To address this problem numerically, we employ an indirect deterministic descent method, leveraging an exact representation of the increment of the objective functional.As a main contribution, we delve into practical aspects in the implementation of the proposed method and expose several results of numerical experiments.

We study the problem of designing “robust” external excitations for control and synchronization of an assembly of homotypic harmonic oscillators representing so-called theta neurons. The model of theta neuron (Theta model) captures, in main, the bursting behavior of spiking cells in the brain of biological beings, enduring periodic oscillations of the electric potential in their membrane. Our task is to find an external stimulus (control), which steers all neurons of a given population to their desired phases (i.e., excites/slows down its spiking activity) with the highest probability. Our methodology is the following: The optimization problem at hand is formulated as an optimal mean-field control problem for the local continuity equation in the space of probability measures. To solve this problem numerically, we propose an indirect deterministic descent method based on an exact representation of the increment (infinite-order variation) of the objective functional. We illustrate the modus operandi of the proposed method discuss some aspects of its practical realization and provide some results of numerical experiments.

We consider the following time-optimal control problem with state constraints: compute minimal travelling time of a controllable object moving in a prescribed flow field in a bounded domain between two given points. The optimal control problem is solved numerically using two direct methods - interior-point line search filter method and sequential quadratic programming. Five sample flows are considered, and computational properties of the corresponding simulations are measured and discussed.

This paper presents a semidiscrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot, and Sarti. We propose a new ingredient, namely, working on the group of translations and discrete rotations $SE(2,N)$. The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of $SE(2,N)$. Harmonic analysis over this group leads to very simple finite dimensional reductions. We then apply these ideas to the inpainting problem which is reduced to the integration of a completely parallelizable finite set of Mathieu-type diffusions (indexed by the dual of $SE(2,N)$ in place of the points of the Fourier plane, which is a drastic reduction). The integration of the the Mathieu equations can be performed by standard numerical methods for elliptic diffusions and leads to a very simple and efficient class of inpainting algorithms. We illustrate the performances of the method on a series of deeply corrupted images.